C7

1
C7 Asymptotes of Rational and Other Functions

• IB Math HL/SL - Santowski

2
(A) Introduction

• To help make sense of any of the following
  discussions, graph all equations and view the
  resultant graphs as we discuss the concepts
• It may also be helpful to use the graphing
  technology to generate a table of values as you
  view the graphs
• Use either WINPLOT or a GDC

3
(A) Review of Rational Functions

• A rational number is a number that can be written
  in the form of a fraction. So likewise, a
  rational function is function that is presented
  in the form of a fraction.
•  We have seen two examples of rational functions
  in this course  ? we can generate a graph of
  polynomials when we divide them in our work with
  the Factor Theorem ? i.e. Q(x) (x3 - 2x
  1)/(x - 1). If x-1 was a factor of P(x), then we
  observed a hole in the graph of Q(x). If x-1 was
  not a factor of P(x), then we observed an
  asymptote in the graph of Q(x)

4
(A) Review of Rational Functions
5
(A) Review of Rational Functions

• We have seen several examples of rational
  functions in this course, when we investigated
  the reciprocal functions of linear fcns, i.e.
  f(x) 1/(x 2) and quadratic fcns, i.e. g(x)
  1/(x2 - 3x - 10) and the tangent function y
  tan(x).

6
(B) Domain, Range, and Zeroes of Rational
Functions

• Given the rational function r(x) n(x)/d(x) ,
• The domain of rational functions involve the fact
  that we cannot divide by zero. Therefore, any
  value of x that creates a zero denominator is a
  domain restriction. Thus in r(x), d(x) cannot
  equal zero.
• For the zeroes of a rational function, we simply
  consider where the numerator is zero (i.e. 0/d(x)
  0). So we try to find out where n(x) 0
• To find the range, we must look at the various
  sections of a rational function graph and look
  for max/min values
•  
• EXAMPLES Graph and find the domain, range,
  zeroes of
• f(x) 7/(x 2),
• g(x) x/(x2 - 3x - 4), and
• h(x) (2x2 x - 3)/(x2 - 4)

7
(B) Domain, Range, and Zeroes of Rational
Functions

• EXAMPLES Graph and find the domain, range,
  zeroes of
• f(x) 7/(x 2),
• g(x) x/(x2 - 3x - 4), and
• h(x) (2x2 x - 3)/(x2 - 4)

8
(C) Vertical and Horizontal Asymptotes

• Illustrate with a graph of y 1/x and draw
  several others (i.e. pg 348)
• A vertical asymptote occurs when the value of the
  function increases or decreases without bound as
  the value of x approaches a from the right and
  from the left.
• We symbolically present this as f(x) ? 8 as x ?
  a or x ? a-
• We re-express this idea in limit notation ? lim x
  ? a f(x) 8
• A horizontal asymptotes occurs when a value of
  the function approaches a number, L, as
  x increases or decreases without bound.
• We symbolically present this as as f(x) ? a as
  x ? 8 or x ? - 8
• We can re-express this idea in limit notation ?
  lim x ? 8 f(x) a

9
(D) Finding the Equations of the Asymptotes

• To find the equation of the vertical asymptotes,
  we simply find the restrictions in the
  denominator and there is our equation of the
  asymptote i.e. x a
• To find the equation of the horizontal
  asymptotes, we can work through it in two
  manners. First, we can prepare a table of values
  and make the x value larger and larger positively
  and negatively and see what function value is
  being approached
• The second approach, is to rearrange the equation
  to make it more obvious as to what happens when
  x gets infinitely positively and negatively.

10
(D) Finding the Equations of the Asymptotes

• ex. Find the asymptotes of y (x2)/(3x-2)
• So we take lim x ? 2/3 f(x) 8 and lim x ?
  2/3- f(x) - 8 ? thus we have an asymptote at
  x 2/3
• To find the horizontal asymptote ? a table of
  values (or simply large values for x) returns the
  following values
• x 109 ? f(109) 0.3333333342 or close to 1/3
• x -(109) ? f(-(109)) 0.3333333342 or close to
  1/3

11
(D) Finding the Equations of the Asymptotes

• Alternatively, we can find the horizontal
  asymptotes of y (x2)/(3x-2) using algebraic
  methods ? divide through by the x term with the
  highest degree
• as x ? 8, then 2/x ? 0

12
(E) Examples

• Further examples to do ? Find vertical and
  horizontal asymptotes for
• y (4x)/(x21)
• y (2-3x2)/(1-x2)
• y (x2 - 3)/(x5)

13
(E) Examples

• y (4x)/(x21)
• y (2-3x2)/(1-x2)
• y (x2 - 3)/(x5)

14
(F) Graphing Rational Functions

• If we want to graph rational functions (without
  graphing technology), we must find out some
  critical information about the rational function.
  If we could find the asymptotes, the domain and
  the intercepts, we could get a sketch of the
  graph
• ex gt f(x) (x2)/(x3-2x2 - 5x 6)
•  
• NOTE after finding the asymptotes (at x -2,
  1,3) we find the behaviour of the fcn on the left
  and the right of these asymptotes by considering
  the sign of the 8 of f(x).

15
(F) Oblique Asymptotes

• Some asymptotes that are neither vertical or
  horizontal gt they are slanted. These slanted
  asymptotes are called oblique asymptotes.
•  
• Ex. Graph the function f(x) (x2 - x - 6)/(x -
  2) (which brings us back to our previous work on
  the Factor Theorem and polynomial division)
• Recall, that we can do the division and rewrite
  f(x) (x2 - x - 6)/(x - 2) as f(x) x 1 -
  4/(x - 2).
• Again, all we have done is a simple algebraic
  manipulation to present the original equation in
  another form.
• So now, as x becomes infinitely large (positive
  or negative), the term 4/(x - 2) becomes
  negligible i.e. 0.
• So we are left with the expression y x 1 as
  the equation of the oblique asymptote.

16
(G) Internet Links

• Rational Functions from WTAMU
• Calculus_at_UTK 2.5 - Limits Involving Infinity
• Calculus I (Math 2413) - Limits - Limits
  Involving Infinity from Paul Dawkins
• Limits Involving Infinity from P.K. Ving

17
(G) Homework

•  
• MCB4U
• DAY 1 Nelson text, p356, Q1-4
• DAY 2 Nelson text, p357, Q10,11,12,14,15
• IB Math HL/SL
• Stewart, 1989, Chap 5.1, p212, Q2,3
• Stewart, 1989, Chap 5.2, p222, Q2-6
• Stewart, 1989, Chap 5.6, p244, Q1,2